Vayne has one of the weakest laning phases of the carries. This is a trade off for being one hell of a monster late game.
Blitzcrank is not a good combo with Vayne simply because you will probably be beat by any other ADC + Support combination even if Blitz gets a pull off. You want someone that can defend, poke or peel for you, like Sona (for poke to keep your opponents off you, and healing), Taric (again, for healing and stun to protect you), or Cow (for protection).
One thing you will have to learn if you want to play Vayne is how to farm under your tower. You will be pushed to your tower by any competent bot lane so you have to be able to make due with whatever CS you can grab. You have to expect a very one-sided laning phase as Vayne - you will be losing no matter what. You just have to mitigate how much you lose by.
If you can come out even with opposing carry as Vayne, you've already won your game.